Ray Shooting Data Structures

• Ray shooting data structures are computational frameworks that quickly determine the first intersection between a ray and a set of geometric objects.
• They utilize methods like segment trees, fractional cascading, and polynomial partitioning to optimize query performance across varied dimensions.
• These structures are pivotal in computer graphics, simulations, and optimization, balancing query speed, space efficiency, and dynamic updates.

A ray shooting data structure is a computational framework designed to efficiently answer queries of the form: given a set of geometric objects, and a query ray (half-line or full line), report the first object (e.g., segment, triangle, arc, voxel, or mesh cell) intersected by the ray. These structures are central to numerous domains—including computational geometry, computer graphics, scientific simulation, and optimization—where spatial queries must be performed rapidly over complex or dynamic scenes. Ray shooting problems motivate a rich spectrum of algorithmic and data-structural techniques, ranging from planar segment trees and sublinear hull structures to high-dimensional partitioning and compressed hardware-amenable formats.

1. Foundational Principles of Ray Shooting Data Structures

At the core, ray shooting addresses the problem of determining, for each query ray $\rho$, the earliest intersection with a member of a predefined set $S$ of geometric primitives. This intersection can be with segments (planar), polygons, polyhedral faces (3D), algebraic arcs, volumetric voxels, or more complex mesh elements. The primitive set $S$ may be static or dynamic; the query $\rho$ may be axis-aligned, arbitrary, or restricted. Data structures differ in space–query–update trade-offs and in how efficiently they address intersection locality, object complexity, dimensionality, and query patterns.

Key properties:

• Intersection Detection: Fast determination of first hit.
• Query Locality: Efficient traversal using spatial or partitioning schemes.
• Dynamic Operations: Support for insertion and deletion of primitives.
• Space Complexity: Optimization to store $S$ compactly, possibly with empirical or theoretical sublinear scaling.

Major design techniques include axis-aligned grids, hierarchical trees (segment, BVH, Kd-tree), discrete geometric partitioning (polynomial partitions, Voronoi, DAGs), and algorithmic optimizations such as fractional cascading, dynamic traversal, and streamlining in hardware-oriented contexts (Biedl et al., 8 Jan 2026, Ezra et al., 2021, Wang, 2020, Chan et al., 2024).

2. Planar and Axis-Aligned Ray Shooting: Segment-based Structures

In planar contexts, such as intersection queries among line segments or intervals, axis-aligned segment trees and interval trees form the canonical base. The seminal dynamic orthogonal ray-shooting structure by Giyora and Kaplan employs segment trees with dynamic fractional cascading and van Emde Boas shims to maintain sets of disjoint segments—$S_H$ (horizontal) and $S_V$ (vertical)—supporting $O(\log n)$ insert, delete, and first-hit ray shooting queries. This is utilized in algorithmic problems like $O(n\log n)$ minimal dominating set computation on RDV graphs, where operations traverse priority search trees and invoke downward or rightward axis-aligned rays to efficiently identify and mark dominated segments (Biedl et al., 8 Jan 2026).

Such planar structures typically maintain:

• Axis-aligned, disjoint segment sets: $S_H$, $S_V$.
• Operations: insert($s$), delete($s$), orthrayshoot($\rho$) in $O(\log n)$ time.
• Space: $O(n)$ overall, given endpoint universe $\{1,\dots,O(n)\}$.
• No bit tricks needed (integer coordinate assignment suffices).

The segment tree is also employed in the subpath convex hull and ray shooting among planar segments: a subpath-hull engine preprocesses the path $\pi$, supporting $O(\log n)$ hull queries, and an embedding in the segment tree, combined with percolate-up/down query phases, yields $O(\sqrt{n}\log n)$ query time with $O(n\log n)$ space. In the case of nonintersecting segments or full lines, space drops to $O(n)$ (Wang, 2020).

3. High-Dimensional Ray Shooting: Polynomial Partitioning and Triangles

For problems in three dimensions or involving algebraic curves, polynomial partitioning methods—coming from the Guth-Katz theorem—enable sublinear query structures for ray shooting among triangles or algebraic arcs. The approach recursively partitions space with a constant-degree polynomial $f(x,y,z)$, generating cells with bounded triangle-edge crossings. In each cell, triangles are classified as "narrow" (edges cross cell) or "wide" (interior, not crossing cell boundary); the recursion on narrow triangles achieves rapid size decay, while wide triangles are handled by local "within-cell" structures.

The 3D triangle ray-shooting structure achieves:

• Space and preprocessing: $O(n^{3/2+\varepsilon})$ for $n$ triangles.
• Query: $O(n^{1/2+\varepsilon})$ for general rays (Ezra et al., 2021).
• Improved storage–query tradeoffs: With space $s$ parameterized, query time $O(n^{5/4+\varepsilon}/s^{1/2})$ for $s=O(n^{3/2+\varepsilon})$ and $O(n^{4/5+\varepsilon}/s^{1/5})$ for superlinear $s$.
• For $m$ queries, running time attains $$\max\{O(m^{2/3}n^{5/6+\varepsilon} + n^{1+\varepsilon}), O(m^{5/6+\varepsilon}n^{2/3} + m^{1+\varepsilon})\}$$.

In 2D algebraic contexts, similar polynomial partitioning yields $O(n^{3/2+\varepsilon})$ space and $O(n^{1/4+\varepsilon})$ query for ray shooting amid Jordan arcs (Chan et al., 2024).

4. Voxel, BVH, and Mesh-based Ray Shooting in Graphics and Simulation

Applications in rendering, volume path tracing, and scientific simulation motivate data structures that are hardware-efficient and numerically robust. GPU-oriented ray shooting employs hybrid voxel formats, compressed wide BVHs, and block-structured AMR for rapid culling and traversal.

Hybrid Voxel Formats

Hybrid voxel hierarchies compose levels of different base formats—raw grid, distance field, sparse voxel octree/DAG—exposing a Pareto frontier of memory vs intersection performance (Arbore et al., 2024). Construction proceeds bottom-up with Morton order for out-of-core builds; traversal is recursive, per-level, using fast DDA stepping and intersection kernels. Storage is parameterized by block size and format; expected traversal steps are minimized with high sparsity and deduplication.

BVH Compression and Fixed-point Ray Streams

Compressed BVH structures store node bounds and leaf triangles in 8-bit quantized local frames, using power-of-two scaling per axis. Fixed-point arithmetic ensures watertightness and prevents geometric holes.

• BVH nodes: 96 bytes in 8-wide compressed nodes vs 228 bytes uncompressed.
• Triangle storage: 9 bytes per quantized triangle.
• Ray stream tracing: dynamically bind rays to BVH nodes, process bulk intersections with fixed-point box and triangle routines.
• Memory traffic reduction: quantized 8-wide BVH stream achieves traffic $18\%$ of traditional approaches; triangles and nodes are never decompressed (Grauer et al., 30 May 2025).

Subspace Culling for Ray-Box Intersection

Per-AABB voxel masks (e.g., $R=4$ resolution) and ray masks (from LUT-encoded voxel entries/exits) are intersected by bitwise AND. This culling method yields significant reductions in false-positive intersections (up to $50\%$ fewer intersection tests), with mask compression up to $27\times$ via LUTs (Yoshimura et al., 2023).

AMR and Grid/DDA in Volume Rendering

Volume path tracing of AMR datasets leverages grid+DDA traversal, which is shown to outperform primary-ray BVH approaches for incoherent workloads. Fine-grained grids provide tighter majorants and lower memory footprints. The grid size must be tuned to data scale; interactive transfer function updates are supported in grid-based structures, while static transfer functions may benefit from preclassified kd-tree bounds (Zellmann et al., 2022).

5. Unstructured Cell-based and Tetrahedral Mesh Ray Shooting

Ray shooting in unstructured mesh contexts—such as tetrahedralizations—is achieved by space-filling reordering and XOR-based compact cell representations. The traversal algorithm projects the ray into a locally orthonormal basis, leverages rapid 2D determinant orientation tests to decide exit faces, and recovers missing vertices and neighbors per cell using XOR aggregation. These techniques yield:

• Storage: Tet32 (32 bytes), Tet20 (20 bytes), Tet16 (16 bytes); substantial compression over classic mesh formats.
• Traversal: $O(1)$ work per tetrahedron; cache locality enhanced by Hilbert curve ordering.
• GPU-implementation: occupancy and register pressure optimized; kernel times competitive with BVH and k-d reference structures (Aman et al., 2021).

6. Algorithmic, Complexity-theoretic, and Implementation Aspects

Ray shooting structures' complexity varies with the object set, query pattern, dimensionality, and dynamic requirements. Key results include:

Structure Type	Space	Preprocessing	Query (worst-case)
Planar, axis-aligned	$O(n)$	$O(n)$	$O(\log n)$
Planar, general segs	$O(n\log n)$	$O(n^{1.5})$	$O(\sqrt{n}\log n)$
3D triangles	$O(n^{3/2+\varepsilon})$	$O(n^{3/2+\varepsilon})$	$O(n^{1/2+\varepsilon})$
Algebraic arcs (2D)	$O(n^{3/2+\varepsilon})$	$O(n^{3/2+\varepsilon})$	$O(n^{1/4+\varepsilon})$
Voxel/BVH hybrid	$O(\sum N_\ell S(F_\ell))$	(varies)	(empirical, $O(1)$ per node)

Algorithmic implementations use priority search trees, segment/interval trees, block-list acceleration, partition trees, Morton/space-filling order, bitwise LUT compression, fixed-point arithmetic, and stream-based traversal. Hardware implementation notes emphasize register usage, stack occupancy, and integer/fixed-point arithmetic for predictability and performance.

7. Applications, Impact, and Future Directions

Ray shooting data structures underpin a variety of computational domains:

• Geometric optimization and combinatorics: dominating sets in graphs, intersection reporting, arrangement construction, convex hull queries.
• Computer graphics: acceleration of ray tracing and path tracing, voxel rendering, memory-bound hardware rasterization.
• Scientific computing: radiative transfer simulation on unstructured grids (Voronoi, tetrahedral), fluid dynamics postprocessing (AMR).
• Hardware ray tracing: direct mapping of quantized BVH/triangle structures to GPU/FPGA pipelines, cloud-rendering in bandwidth-constrained settings (Grauer et al., 30 May 2025, Yoshimura et al., 2023, Zellmann et al., 2022, Aman et al., 2021, Arbore et al., 2024).

Research trends continue to explore improved partitioning strategies, multi-level hybrid formats, stream/packet tracing, compressed representations, and sublinear query algorithms. There is clear movement toward supporting complex objects (algebraic arcs, mesh cells), minimizing space for massive datasets, and adapting structures for next-generation hardware. A plausible implication is that further advances in polynomial partitioning and hybrid hierarchical representations will yield even lower space/query exponents for high-dimensional, high-complexity geometric datasets.